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There is a continuum of consumers indexed by θ, with θ ∈ [0, θ]. The index
θ is assumed to follow a uniform distribution, and represents the consumers’
tastes for the quality of the product. We assume that there is no resale market
for used product. Each consumer is assumed to buy at most a unit of the
product.
The utility of a type θ consumer is,
⎧
⎨ θq i − p i if the consumer buys the original product
U(θ) = θq p − p p if the consumer buys the pirated product (1)
⎩
0 if the consumer does not buy
where p i , q i , p p and q p are the price and quality of the original and the pirated
product, respectively. We assume q i > q p > 0. Let r = q i /q p > 1 be the ratio
of qualities.
Since qualities are common knowledge, decisions on prices are equivalent to
decisions on quality-deflated prices. So, we consider that the incumbent and the
pirate decide on hedonic prices x i = p i /q i and x p = p p /q p respectively. Without
loss of generality, we also assume that x i , x p ∈ [0, ¯θ]. 8
Demand functions are obtained from the indifferent consumers according to
expression (1). When the pirate is not in the market, consumers only observe
x i and the incumbent faces the monopoly demand:
D i (x i ) = θ − x i , (2)
where θ i = x i is the consumer who is indifferent between buying the original
product and not buying. Of course, the pirate’s demand is D p ≡ 0 in this case,
which is trivial. Thus, the analysis focus on the duopolistic setting.
If the pirate is in the market, consumers consider both offers, the legal and
the illegal one, and three kinds of indifferent consumers may exist. First, if
there is a consumer who is indifferent between buying the original and the
pirated product, he must be θ o = (p i − p p ) / (q i − q p ), or, equivalently, θ o =
(rx i − x p )/(r − 1). Second, if there is a consumer who is indifferent between
buying from the pirate and not buying at all, he must be θ p = x p . Third, there
may be a consumer who is indifferent between buying the original product and
not buying at all, when the pirate fixes a very high price. This consumer must be
θ i = x i . Some easy calculations show that the demands faced by the incumbent
and the pirate are
8 Our technical specification is similar to Ronnen’s (1991) approach, in which hedonic prices
are considered instead of usual prices. Of course, our analysis can be equivalently carried out
without hedonic prices. The best response prices that we obtain are obviously similar to those
in Ronnen (1991).
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